On Symbolic Scheduling Independent Tasks with Restricted Execution Times

Abstract

Ordered Binary Decision Diagrams (OBDDs) are a data structure for Boolean functions which supports many useful operations. It finds applications in CAD, model checking, and symbolic graph algorithms. We present an application of OBDDs to the problem of scheduling N independent tasks with k different execution times on m identical parallel machines while minimizing the over-all finishing time. In fact, we consider the decision problem if there is a schedule with makespan D. Leung’s dynamic programming algorithm solves this problem in time ${\mathcal O}({\rm log} m \cdot N^{2(k-1)})$. In this paper, a symbolic version of Leung’s algorithm is presented which uses OBDDs to represent the dynamic programming table T. This heuristical approach solves the scheduling problem by executing ${\mathcal O}(k {\rm log} m {\rm log}(mD))$ operations on OBDDs and is expected to use less time and space than Leung’s algorithm if T is large but well-structured. The only known upper bound of ${\mathcal O}((m \cdot D)^{3k+2})$ on its resource usage is trivial. Therefore, we report on experimental studies in which the symbolic method was applied to random scheduling problem instances.